Let $f(x) = -6x^{2}-4x+6$. Where does this function intersect the x-axis (i.e. what are the roots or zeroes of $f(x)$ )?
Solution: The function intersects the x-axis when $f(x) = 0$ , so you need to solve the equation: $-6x^{2}-4x+6 = 0$ Use the quadratic formula to solve $ax^2 + bx + c = 0$ $x = \dfrac{-b \pm \sqrt{b^2 - 4ac}}{2a}$ $a = -6, b = -4, c = 6$ $ x = \dfrac{+ 4 \pm \sqrt{(-4)^{2} - 4 \cdot -6 \cdot 6}}{2 \cdot -6}$ $ x = \dfrac{4 \pm \sqrt{160}}{-12}$ $ x = \dfrac{4 \pm 4\sqrt{10}}{-12}$ $x =\dfrac{1 \pm \sqrt{10}}{-3}$